I would start with just a few different things, otherwise you could end up with nothing.
If I had very limited space, I wonder what I would grow

I think I would always grow a few potatoes as they are so magical to dig up, and homegrown new potatoes are on a different plane to ones which have sat in a sack in the shop for a week. They are also fairly foolproof - well, apart from frost and blight and scab and blackleg and ............. But mostly they are very easy, especially in your first year.
I would also always grow Sugar Snap peas, where you eat the whole pod once it's swollen with peas (unlike mangetout where you eat the pods while still flat) Go for a dwarf one such as Quartz, as some varieties are many feet tall, so require complicated support.
I grow ordinary podding peas, but only enough to graze on whilst working in the garden, or to use fresh in salads - too much of a fandango to prepare for several platesful.
I always grow some beans of some sort. Broad beans are hardy and very easy to grow but are an acquired taste. Runner beans are always described as easy, but where I live (high, cold and draughty) they are hard to grow and I have them inside my polytunnel. French beans are probably the easiest, and the dwarf type probably simpler for your first year. They, like runner beans, are frost tender, so don't sow them when the soil is still cold. They grow and crop quite quickly so can safely be sown in early June.
I love a few carrots to eat raw, but concentrate mainly on a large variety of salad crops, salad leaves and lettuce, cucumbers (not really for your first year or without cold protection), peppers and chillies (under protection or indoors), beetroot, mizuna and so on. Tomatoes need protection in most of Britain, but if you want some choose a variety which claims to do well outdoors. As well as my tunnel variety (Sakura) I am growing one - can't remember it's name just now but will modify this if I find the packet - which is recommended for containers and specifically for outdoors

You can start off tomatoes indoors on a windowsill, then plant them out after the last frost.
I would always grow some winter brassicas, for me purple sprouting broccoli and kale are indispensable - they can be sown from late April to mid May to give them time to grow to a decent size before winter gets here, but they are winter hardy.
You can grow herbs in pots both indoors and in any small sunny space in the garden - they lift a plain salad to something special and are wonderful in cooking.
Those are my essentials for starters.